We now define a map \(\widehat{\mathbf {\theta }}^T(t, \mathbf {\mu })\)  for any time \(t\)  in the time interval \(\mathbb {T}\)  and any value of the parameter \(\mathbf {\mu }\)  by training artificial neural networks (ANN) to approximate \(\mathbf {\theta }^T(t, \mathbf {\mu })\)  based on the training data points obtained by \(L^2\)  projection [1]}. The neural networks are built on the PyTorch platform [2]}. For simplicity, our ANN has five hidden layers, and each layer has seven neurons. We use tanh as our activation function. A detailed discussion on these hyperparameters could be found in [3]}. Here, we use a mean squared error (\(\mathrm {MSE}^{\theta , T}\) ) as the metric of our network loss function, defined as follows
\({\mathrm {MSE}^{\theta , T}}=\frac{1}{\mathrm {M} N^t} \sum _{i=1}^{\mathrm {M}}\sum _{k=0}^{N^t}\left|\widehat{\mathbf {\theta }}^T\left(t^k, \mathbf {\mu }^{(i)}\right)-{\mathbf {\theta }^T}\left(t^k, \mathbf {\mu }^{(i)}\right)\right|^{2}.\) 
